On the Broadband Synchrotron Spectra of Pulsar Wind Nebulae

To facilitate the transition from the Poynting-flux-dominated (i.e., high-σ) PW (Arons 2002) to the particle energy flux-dominated (i.e., low-σ) downstream of the termination shock (Rees & Gunn 1974; Kennel & Coroniti 1984; Begelman & Li 1992; Bogovalov et al. 2005), efficient dissipative processes are necessary to convert the magnetic energy to kinetic energy in the upstream wind or at the termination shock. Magnetic reconnection serves as the required dissipation mechanism (Del Zanna et al. 2016). It can occur in the striped wind with alternating magnetic fields (Coroniti 1990; Lyubarsky & Kirk 2001), as well as the kink-unstable polar jet (Begelman 1998; Pavlov et al. 2003; Mizuno et al. 2011; Mignone et al. 2013).

In resistive MHD theory, the magnetic reconnection depends on the slow resistive diffusion of magnetic fields (Sweet 1958; Parker 1957). In the presence of turbulence, turbulent velocities dominate the magnetic field dynamics in the direction perpendicular to the local magnetic field (LV99). The turbulent velocities act to increase the magnetic field gradients along the turbulent energy cascade toward smaller scales, and thus enhance the reconnection and determine the actual reconnection efficiency. In strongly magnetized turbulence, the reconnection rate is given by (LV99)

$$\begin{eqnarray}&&{v}_{\mathrm{rec}}\approx {V}_{{\rm{A}}}\sqrt{\displaystyle \frac{{L}_{i}}{{L}_{x}}}{M}_{{\rm{A}}}^{2},\end{eqnarray} \tag{ 1 }$$

when the injection scale of turbulence L_i is smaller than the length of the current sheet L_x, and

$$\begin{eqnarray}&&{v}_{\mathrm{rec}}\approx {V}_{{\rm{A}}}\sqrt{\displaystyle \frac{{L}_{x}}{{L}_{i}}}{M}_{{\rm{A}}}^{2}\end{eqnarray} \tag{ 2 }$$

when ${L}_{i}\gt {L}_{x}$. Here ${v}_{\mathrm{rec}}$ is the reconnection speed, V_A is the Alfvén speed, and M_A is the ratio of the turbulent speed at L_i to V_A. Without the constraint imposed by the microscopic diffusion process, turbulent reconnection can be highly efficient with ${v}_{\mathrm{rec}}$ up to V_A. Numerical experiments in both nonrelativistic (Kowal et al. 2009) and relativistic (East et al. 2015; Takamoto et al. 2015; Zrake & East 2016) plasmas confirm that the turbulent reconnection is universal with respect to microscopic physics, e.g., resistive diffusion.

Magnetic reconnection itself acts as a source of turbulence. The magnetostatic free energy carried by the striped wind can be readily converted to turbulent energy via magnetic reconnection (East et al. 2015; Zrake & East 2016). The generated turbulence in turn boosts the reconnection efficiency as discussed above (Kowal et al. 2017). This positive feedback can result in a runaway reconnection and explosive release of magnetic energy, accounting for the energy source of observed γ-ray flares from, e.g., the Crab Nebula (Abdo et al. 2011; Tavani et al. 2011). The development of both turbulence and turbulent reconnection of magnetic fields have been observed in numerical simulations of the pulsar striped wind, e.g., Zrake (2016).

Moreover, magnetic reconnection can also efficiently energize particles (de Gouveia dal Pino & Lazarian 2005; Kowal et al. 2011, 2012; del Valle et al. 2016), which supplies the PWN with a power-law population of energetic electrons. By adopting a simple treatment analogous to that for the first-order Fermi acceleration at shocks, de Gouveia dal Pino & Lazarian (2005) derived a power-law index of 2.5 for the reconnection-accelerated particles. Beyond the termination shock, the spectrum of the accelerated electrons injected into the PWN will be affected by both the ASA and synchrotron cooling in the turbulent magnetic fields downstream.

Due to the magnetic reconnection in the striped wind, the PWN is expected to be fully turbulent. The Kolmogorov-like magnetic energy spectrum has been found in 3D relativistic MHD simulations of PWNe, e.g., Porth et al. (2014b), which is a typical spectral form of turbulent magnetic fields in strong MHD turbulence (Maron & Goldreich 2001; Cho & Lazarian 2003).

In strong MHD turbulence with increased turbulence anisotropy toward smaller scales, the turbulent eddies at the resonant scale of gyroresonance are highly elongated along the magnetic field. With the perpendicular scale of the turbulent eddies much smaller than the parallel resonant scale, particles interact with many uncorrelated turbulent eddies within one gyro orbit, and thus the gyroresonance scattering is inefficient (Chandran 2000; Yan & Lazarian 2002). In the absence of efficient scattering, relativistic particles can slide along magnetic field lines. Obviously, the magnetic field variation is slow with respect to the motion of particles. Within the traps of turbulent eddies, the second (longitudinal) adiabatic invariant of the particles bouncing between the "mirror points" applies. As there are statistically distributed reconnection and dynamo regions in strong MHD turbulence (Xu & Lazarian 2016), after cycles of acceleration in turbulent reconnection regions with shrinking field lines and cycles of deceleration in turbulent dynamo regions with stretched field lines, the particles have a diffusive energy gain (Brunetti & Lazarian 2016). We term this acceleration process as the "ASA" (Xu & Zhang 2017; Xu et al. 2018).

The acceleration rate of the ASA is

$$\begin{eqnarray}&&{a}_{{\rm{A}}}=\xi \displaystyle \frac{{u}_{\mathrm{tur}}}{{l}_{\mathrm{tur}}},\end{eqnarray} \tag{ 3 }$$

where l_tur is the characteristic scale of turbulent magnetic fields, u_tur is the turbulent speed at l_tur, and their ratio gives the corresponding turbulent eddy-turnover time τ_tur. ξ represents the cumulative fractional energy change during τ_tur. As relativistic particles undergo the first-order Fermi process within individual eddies, given the number of bounces $\sim c/{u}_{\mathrm{tur}}$ and the energy change per bounce $\sim {u}_{\mathrm{tur}}/c$, where c is the speed of light, we can easily see that ξ is of order unity. Thus the ASA is a very efficient stochastic acceleration mechanism compared to the second-order Fermi acceleration associated with gyroresonance scattering.

The stochastic acceleration generally leads to energy diffusion and a hard energy spectrum of particles. We consider a steady injection of a power-law distribution of electrons originating from the magnetic reconnection sites in the PW (Section 2.1),

$$\begin{eqnarray}&&Q(E)={{CE}}^{-p},\,\,{E}_{l}\lt E\lt {E}_{u}.\end{eqnarray} \tag{ 4 }$$

At a given reference energy, C is the electron number per unit energy per unit time, p is the power-law index, and E_l and E_u are the lower and upper energy limits, respectively. Due to the ASA, the energy distribution of electrons spreads out in energy space following (Melrose 1969)

$$\begin{eqnarray}&&E\sim \exp (\pm 2\sqrt{{a}_{{\rm{A}}}t}).\end{eqnarray} \tag{ 5 }$$

The injected energy spectrum evolves with time t and approaches a universal form (Xu et al. 2018)

$$\begin{eqnarray}\begin{array}{rcl}N(E,t) & = & \displaystyle \frac{C({E}_{u}^{-p+1}-{E}_{l}^{-p+1})\sqrt{t}}{(-p+1)\sqrt{\pi {a}_{{\rm{A}}}}}{E}^{-1}\exp \left(-\displaystyle \frac{E}{{E}_{\mathrm{cf}}}\right),\\ {E}_{m} & \lt & E\lt {E}_{\mathrm{cf}}.\end{array}\end{eqnarray} \tag{ 6 }$$

The resulting energy spectrum has a minimum energy E_m. It is smaller than E_l as the energy spectrum under the effect of ASA tends to have a larger energy spread than that of the injected electrons (Equation (5)). E_cf is the cutoff energy of the ASA and has the expression

$$\begin{eqnarray}&&{E}_{\mathrm{cf}}=\displaystyle \frac{{a}_{{\rm{A}}}}{\beta },\end{eqnarray} \tag{ 7 }$$

corresponding to the equalization between a_A and synchrotron cooling rate. Here the parameter β is given by

$$\begin{eqnarray}&&\beta =\displaystyle \frac{{P}_{\mathrm{syn}}}{{E}^{2}}=\displaystyle \frac{{\sigma }_{{\rm{T}}}{{cB}}^{2}}{6\pi {\left({m}_{e}{c}^{2}\right)}^{2}},\end{eqnarray} \tag{ 8 }$$

where ${P}_{\mathrm{syn}}$ is the power of synchrotron radiation, B is the magnetic field strength, ${\sigma }_{{\rm{T}}}$ is the Thomson cross section, and m_e is the electron rest mass. Different from a steep energy spectrum of electrons, for which the minimum energy is the characteristic electron energy, for a hard electron spectrum in Equation (6), the upper cutoff energy E_cf dominates the integral,

$$\begin{eqnarray}&&{\int }_{{E}_{m}}^{{E}_{\mathrm{cf}}}{EN}(E){dE}\approx \displaystyle \frac{C({E}_{u}^{-p+1}-{E}_{l}^{-p+1})\sqrt{t}}{(-p+1)\sqrt{\pi {a}_{{\rm{A}}}}}{E}_{\mathrm{cf}}=\epsilon \dot{E}t,\end{eqnarray} \tag{ 9 }$$

where we assume ${E}_{u}\lt {E}_{\mathrm{cf}}$ and ${E}_{m}\ll {E}_{\mathrm{cf}}$, $\dot{E}$ is the spin-down power of the pulsar, and is the fraction of the spin-down energy converted to the particle energy. To confront the above relation to observations, a detailed modeling of the particle acceleration in the PW is needed, which is beyond the scope of the current paper.

We next consider that the injected electrons have sufficiently high energies with ${E}_{\mathrm{cf}}\lt {E}_{u}$. The hard electron energy distribution below E_cf is attributed to the ASA, while the distribution at energies above E_cf is governed by synchrotron cooling effect. By defining the critical cooling energy (Sari et al. 1998)

$$\begin{eqnarray}&&{E}_{c}={P}_{\mathrm{syn}}t=\displaystyle \frac{1}{\beta t},\end{eqnarray} \tag{ 10 }$$

we see that the injected electrons with $E\gt {E}_{c}\gt {E}_{\mathrm{cf}}$ cool down to E_c within time t. But the electrons with $E\lt {E}_{\mathrm{cf}}$ do not cool because of the ASA, irrespective of the value of E_c. It means that the overall system is always in the slow cooling regime.

Given E_m and E_u as the lower and upper energy limits of the entire electron energy spectrum, depending on the relation between E_l, E_cf, and E_c, the electron spectrum at $E\gt {E}_{\mathrm{cf}}$ can exhibit different forms (Xu et al. 2018).

Case (i): ${E}_{c}\lt {E}_{l}\lt {E}_{\mathrm{cf}}$

In the energy range above E_cf, the cooled electrons follow a steep spectrum (Kardashev 1962)

$$\begin{eqnarray}&&N(E)=\displaystyle \frac{{{CE}}^{-(p+1)}}{\beta (p-1)}.\end{eqnarray} \tag{ 11 }$$

Combining Equations (6) and (11), we illustrate the asymptotic behavior of the electron spectrum over a broad energy range in Figure 1(a). The corresponding synchrotron spectrum is

$$\begin{eqnarray}&&\nu {F}_{\nu }=\nu {F}_{\nu ,\max }\propto \nu ,\,\,\nu \lt {\nu }_{\mathrm{cf}},\end{eqnarray} \tag{ 12a }$$

$$\begin{eqnarray}&&\nu {F}_{\nu }=\nu {F}_{\nu ,\max }{\left(\displaystyle \frac{\nu }{{\nu }_{\mathrm{cf}}}\right)}^{-\tfrac{p}{2}}\propto {\nu }^{-\tfrac{p}{2}+1},\,\,\,\nu \gt {\nu }_{\mathrm{cf}},\end{eqnarray} \tag{ 12b }$$

as displayed in Figure 1(b), where ${\nu }_{\mathrm{cf}}$ is the frequency related to E_cf, F_ν is the flux density at frequency ν, and ${F}_{\nu ,\max }$ is the maximum value of F_ν.

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Case (ii): ${E}_{c}\lt {E}_{\mathrm{cf}}\lt {E}_{l}$

In the energy range above E_l, the same steep electron spectrum as in Equation (11) applies. Moreover, driven by the cooling effect, the injected electron energy distribution extends down to lower energies. In the energy range ${E}_{\mathrm{cf}}\lt E\lt {E}_{l}$, the electron spectrum is $N(E)\sim {E}^{-2}$, with the injection and radiation losses of electrons in equilibrium (Melrose 1969). Accordingly, the entire electron spectrum contains three segments, as shown in Figure 1(c). The corresponding synchrotron spectrum is (see Figure 1(d)),

$$\begin{eqnarray}&&\nu {F}_{\nu }=\nu {F}_{\nu ,\max }\propto \nu ,\,\,\nu \lt {\nu }_{\mathrm{cf}},\end{eqnarray} \tag{ 13a }$$

$$\begin{eqnarray}&&\nu {F}_{\nu }=\nu {F}_{\nu ,\max }{\left(\displaystyle \frac{\nu }{{\nu }_{\mathrm{cf}}}\right)}^{-\tfrac{1}{2}}\propto {\nu }^{\tfrac{1}{2}},\,\,\,{\nu }_{\mathrm{cf}}\lt \nu \lt {\nu }_{l},\end{eqnarray} \tag{ 13b }$$

$$\begin{eqnarray}&&\begin{array}{l}\nu {F}_{\nu }=\nu {F}_{\nu ,\max }{\left(\displaystyle \frac{{\nu }_{l}}{{\nu }_{\mathrm{cf}}}\right)}^{-\tfrac{1}{2}}{\left(\displaystyle \frac{\nu }{{\nu }_{l}}\right)}^{-\tfrac{p}{2}}\propto {\nu }^{-\tfrac{p}{2}+1},\\ \qquad \qquad \qquad \nu \gt {\nu }_{l},\end{array}\end{eqnarray} \tag{ 13c }$$

where ${\nu }_{l}$ is related to E_l.

Case (iii): ${E}_{l}\lt {E}_{c}\lt {E}_{\mathrm{cf}}$

As shown in Figure 2(a), N(E) in this situation has the same form as that in Case (i). The resulting synchrotron spectrum also has the same shape as that in Equation (12), which can be seen in Figure 2(b).

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Case (iv): ${E}_{l}\lt {E}_{\mathrm{cf}}\lt {E}_{c}$

The cooled electrons at $E\gt {E}_{c}$ still follow the spectral shape as in Equation (11), while the electrons in the energy range ${E}_{\mathrm{cf}}\lt E\lt {E}_{c}$ are not subjected to the cooling effect, so their original spectral form remains (see Figure 2(c)). Then the synchrotron spectrum is given by (Figure 1(d)),

$$\begin{eqnarray}&&\nu {F}_{\nu }=\nu {F}_{\nu ,\max }\propto \nu ,\,\,\nu \lt {\nu }_{\mathrm{cf}},\end{eqnarray} \tag{ 14a }$$

$$\begin{eqnarray}&&\begin{array}{l}\nu {F}_{\nu }=\nu {F}_{\nu ,\max }{\left(\displaystyle \frac{\nu }{{\nu }_{\mathrm{cf}}}\right)}^{-\tfrac{p-1}{2}}\propto {\nu }^{-\tfrac{p-3}{2}},\\ \qquad \qquad {\nu }_{\mathrm{cf}}\lt \nu \lt {\nu }_{c},\end{array}\end{eqnarray} \tag{ 14b }$$

$$\begin{eqnarray}&&\begin{array}{l}\nu {F}_{\nu }=\nu {F}_{\nu ,\max }{\left(\displaystyle \frac{{\nu }_{c}}{{\nu }_{\mathrm{cf}}}\right)}^{-\tfrac{p-1}{2}}{\left(\displaystyle \frac{\nu }{{\nu }_{c}}\right)}^{-\tfrac{p}{2}}\propto {\nu }^{-\tfrac{p}{2}+1},\\ \qquad \qquad \nu \gt {\nu }_{c},\end{array}\end{eqnarray} \tag{ 14c }$$

where ${\nu }_{c}$ corresponds to E_c. It has a similar form to that in Case (ii) (Equation (13)) except for the intermediate segment.

In all the above cases, we also expect $\nu {F}_{\nu }\propto {\nu }^{4/3}$ at $\nu \lt {\nu }_{m}$ (not shown in Figures 1 and 2), where ${\nu }_{m}$ corresponds to E_m. This low-frequency power-law index is a characteristic feature of synchrotron radiation and is independent of the detailed electron energy spectrum (Rybicki & Lightman 1979; Meszaros & Rees 1993).

The above analysis shows that the dominance of the ASA at low energies and synchrotron cooling at high energies naturally gives rise to broken power-law electron and synchrotron spectra. Moreover, depending on the relation between E_l, E_cf, and E_c, the spectrum can have various forms at intermediate energies (frequencies).

Figure 3(a) presents the observed multiwavelength spectrum of G359.23-0.82, a.k.a., the Mouse PWN powered by PSR J1747-2958. Over the radio and X-ray data points taken from Klingler et al. (2018; hereafter K18), we first assume the spectral shape given in Equation (12) corresponding to Case (i) or Case (iii) in Section 2.3. Here we adopt p = 2.18 according to the fit to the X-ray data in K18. We see that the two spectral segments dominated by the ASA and radiation losses separately match the radio and X-ray data.

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The photon energy ${E}_{\mathrm{ph}}$ is related to the electron Lorentz factor ${\gamma }_{e}$ by

$$\begin{eqnarray}&&{E}_{\mathrm{ph}}={\hslash }\displaystyle \frac{{eB}}{{m}_{e}c}{\gamma }_{e}^{2}{\rm{\Gamma }},\end{eqnarray} \tag{ 15 }$$

with ${\hslash }=h/2\pi$, the Planck constant h, the electron charge e, the Lorentz factor ${\rm{\Gamma }}(\approx 1)$ of the mildly relativistic bulk flow in the PWN (Reynolds et al. 2017). The lowest observed photon energy ${E}_{\mathrm{ph},{\rm{m}}}=6.2\times {10}^{-7}$ eV provides the upper limit of ${\gamma }_{e,{\rm{m}}}={E}_{m}/({m}_{e}{c}^{2})$,

$$\begin{eqnarray}&&{\gamma }_{e,{\rm{m}},\max }=5.2\times {10}^{2}{\left(\displaystyle \frac{{\rm{\Gamma }}}{1}\right)}^{-\tfrac{1}{2}}{\left(\displaystyle \frac{B}{200\mu {\rm{G}}}\right)}^{-\tfrac{1}{2}}{\left(\displaystyle \frac{{E}_{\mathrm{ph},{\rm{m}}}}{6.2\times {10}^{-7}\mathrm{eV}}\right)}^{\tfrac{1}{2}}.\end{eqnarray} \tag{ 16 }$$

Here we adopt the equipartition magnetic field $B\sim 200\,\mu {\rm{G}}$ estimated in K18.

The spectral break appears at the photon energy ${E}_{\mathrm{ph},{\rm{b}}}\approx 9\times {10}^{-3}$ eV. It is related to the cutoff electron energy ${E}_{\mathrm{cf}}={\gamma }_{e,\mathrm{cf}}{m}_{e}{c}^{2}$ and signifies the balance between acceleration and cooling. So we have

$$\begin{eqnarray}&&{\gamma }_{e,\mathrm{cf}}=6.2\times {10}^{4}{\left(\displaystyle \frac{{\rm{\Gamma }}}{1}\right)}^{-\tfrac{1}{2}}{\left(\displaystyle \frac{B}{200\mu {\rm{G}}}\right)}^{-\tfrac{1}{2}}{\left(\displaystyle \frac{{E}_{\mathrm{ph},{\rm{b}}}}{9\times {10}^{-3}\mathrm{eV}}\right)}^{\tfrac{1}{2}}.\end{eqnarray} \tag{ 17 }$$

It gives the upper limit of the minimum Lorentz factor of injected electrons ${\gamma }_{e,l}={E}_{l}/({m}_{e}{c}^{2})$ in this scenario. It also gives the minimum cooling time for the condition ${E}_{c}\lt {E}_{\mathrm{cf}}$ to be satisfied (Equations (8), (10), and (17)),

$$\begin{eqnarray}\begin{array}{rcl}{t}_{{\rm{c}},\min } & = & \displaystyle \frac{1}{{E}_{\mathrm{cf}}\beta }=\displaystyle \frac{1}{{\gamma }_{e,\mathrm{cf}}{m}_{e}{c}^{2}\beta }\\ & = & 9.9{\left(\displaystyle \frac{{\rm{\Gamma }}}{1}\right)}^{\tfrac{1}{2}}{\left(\displaystyle \frac{B}{200\mu {\rm{G}}}\right)}^{-\tfrac{3}{2}}{\left(\displaystyle \frac{{E}_{\mathrm{ph},{\rm{b}}}}{9\times {10}^{-3}\mathrm{eV}}\right)}^{-\tfrac{1}{2}}\,\mathrm{kyr}.\end{array}\end{eqnarray} \tag{ 18 }$$

Besides, the maximum cooling time ${t}_{{\rm{c}},\max }$ is given by the spin-down age ${t}_{\mathrm{sd}}=25.5$ kyr of the pulsar (K18). It corresponds to the electron Lorentz factor (Equations (8), (10), and (16)),

$$\begin{eqnarray}\begin{array}{rcl}{\gamma }_{e,\mathrm{sd}} & = & \displaystyle \frac{1}{{m}_{e}{c}^{2}\beta {t}_{\mathrm{sd}}}\\ & = & 2.4\times {10}^{4}{\left(\displaystyle \frac{B}{200\mu {\rm{G}}}\right)}^{-2}{\left(\displaystyle \frac{{t}_{\mathrm{sd}}}{25.5\mathrm{kyr}}\right)}^{-1}\gt {\gamma }_{e,{\rm{m}},\max },\end{array}\end{eqnarray} \tag{ 19 }$$

implying that the electrons in the Mouse PWN are in the slow cooling regime with ${E}_{c,\min }\gt {E}_{m,\max }$ (see also K18 for the analysis of the cooling regime).

Moreover, ${t}_{{\rm{c}},\min }$ in Equation (18) also gives the acceleration timescale of the ASA (Equation (7)),

$$\begin{eqnarray}&&{\tau }_{\mathrm{acc}}={a}_{{\rm{A}}}^{-1}={t}_{{\rm{c}},\min }.\end{eqnarray} \tag{ 20 }$$

We note that τ_acc in Case (i) or Case (iii) is shorter than t_c.

Including the tentative infrared point (see Figure 3(b) and K18)⁵ requires the spectral shape given by Equation (13) in Case (ii) or Equation (14) in Case (iv). The spectral breaks at lower and higher energies are ${E}_{\mathrm{ph},{\rm{b}}1}\approx 3.1\times {10}^{-4}$ eV, ${E}_{\mathrm{ph},{\rm{b}}2}\approx 0.15\,\mathrm{eV}$ in Case (ii), and ${E}_{\mathrm{ph},{\rm{b}}1}\approx 6.9\times {10}^{-4}$ eV, ${E}_{\mathrm{ph},{\rm{b}}2}\approx 0.18\,\mathrm{eV}$ in Case (iv). Similar to the above analysis, these break energies can be used to infer the characteristic ${\gamma }_{e}$, ${t}_{c}$, and τ_acc (Equations (15), (18), and (20)), as shown in Table 1. We see that because the minimum ${t}_{c}$ required in Case (ii) is much larger than ${t}_{\mathrm{sd}}$ of the pulsar, Case (ii) is disfavored.

Table 1.  Characteristic Parameters Inferred from the Spectral Breaks of the Mouse PWN

${E}_{\mathrm{ph}}$	${E}_{\mathrm{ph},{\rm{m}}}$	${E}_{\mathrm{ph},{\rm{b}}}$ or ${E}_{\mathrm{ph},{\rm{b}}1}$	${E}_{\mathrm{ph},{\rm{b}}2}$
Case (i) or Case (iii)	${\gamma }_{e,{\rm{m}}}\approx 5.2\times {10}^{2}$	${\gamma }_{e,\mathrm{cf}}={\gamma }_{e,l,\max }\approx 6.2\times {10}^{4}$, ${\tau }_{\mathrm{acc}}={t}_{{\rm{c}},\min }\approx 9.9$ kyr	⋯
Case (ii)		${\gamma }_{e,\mathrm{cf}}\approx 1.1\times {10}^{4}$, ${\tau }_{\mathrm{acc}}={t}_{{\rm{c}},\min }\approx 53.5$ kyr	${\gamma }_{e,l}\approx 2.5\times {10}^{5}$
Case (iv)		${\gamma }_{e,\mathrm{cf}}={\gamma }_{e,l,\max }\approx 1.7\times {10}^{4}$,${\tau }_{\mathrm{acc}}\approx 35.8$ kyr	${t}_{c}\approx 2.2$ kyr

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The above comparisons show that our analytical spectral shapes agree well with the observed multiwavelength spectrum of the Mouse PWN. The ASA and synchrotron cooling in the PWN naturally account for the flatness and steepness of the radio and X-ray spectra (F_ν), respectively. More reliable infrared measurements with, e.g., Hubble Space Telescope (HST), James Webb Space Telescope, and ALMA are needed to determine the detailed spectral shape at intermediate energies and to distinguish between Case (i) (or Case (iii)) and Case (iv).

We next compare our analytical spectral shapes with the synchrotron spectra of some other PWNe to further examine the applicability of our model.

In Figure 4, we display the synchrotron spectra of 3C 58 (Slane et al. 2008), G21.5–0.9 (Salter et al. 1989), G0.9 + 0.1 (Tanaka & Takahara 2011), and N157B (Zhu et al. 2018). We also include the infrared data for 3C 58, but we caution that the observed values are only upper limits (Slane et al. 2008). The X-ray data for G21.5–0.9 and N157B were obtained from Chandra ObsIDs 2873, 3699 3700, 5158, 5159, 5165, 5166, 6070, 6071, 6740, 6741, 8371, 8372 (G21.5–0.9), and 2783 (N157B). The data were reprocessed using the Chandra Interactive Analysis of Observations software version 4.9 and the Chandra Calibration Data Base version 4.7.7. All observations were reprocessed with chandra_repro to apply the latest calibrations. We extracted the spectra using specextract and fitted the power-law spectra using the tbabs model of XSPEC (v12.9.1p, which uses absorption cross-sections from Wilms et al. 2000). For G21.5–0.9 we use a circular $2\buildrel{\,\prime}\over{.} 5$ aperture centered on the pulsar, and excluding the bright point source to the south. For N157B, we use a 53'' × 24'' extraction region approximately coincident with the brightest radio contour shown in Figure 3 of Chen et al. (2006) and free of thermal supernova remnant (SNR) filaments. In both cases we use background regions located outside the extent of the PWNe/SNRs, and restricted the spectral fits to the 0.5–8 keV range. We verified that our results are not noticeably contaminated by thermal SNR emission by restricting the spectral fits to energies above 2 keV and refitting; in both cases the resulting power-law slopes were virtually identical. Our fits to the X-ray data imply p = 1.68 for G21.5–0.9 and p = 2.6 for N157B. For 3C 58 and G0.9+0.1, we use p = 2.4 and p = 2.6 according to the fits to their X-ray data (Slane et al. 2008; Tanaka & Takahara 2011).

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For 3C 58 and G21.5–0.9, as indicated by the data, we apply the spectral shape in Equation (13) (Case (ii)) and that in Equation (12) (Case (i) or Case (iii)), respectively. For G0.9+0.1 and N157B, the measurements are insufficient to constrain the detailed spectral form in the intermediate energy range, so we present all possible spectral shapes for comparison with future observations. In all cases, it is clear that the electrons in the PWN should undergo the stochastic acceleration to account for the radio spectrum. The general agreement between our analytical spectral shapes and observed spectra implies the ASA and synchrotron cooling together as a common origin of broadband synchrotron spectra of PWNe.

The spectral break between radio and X-ray bands or radio and infrared bands is determined by τ_acc of the ASA. When τ_acc is sufficiently short for E_cf to be larger than both E_c and E_l, the synchrotron spectrum of a PWN falls in Case (i) or Case (iii). Then we can rewrite Equation (18) as

$$\begin{eqnarray}&&\left(\displaystyle \frac{{\tau }_{\mathrm{acc}}}{1\,\mathrm{kyr}}\right){\left(\displaystyle \frac{B}{100\mu {\rm{G}}}\right)}^{\tfrac{3}{2}}=26.5{\left(\displaystyle \frac{{\rm{\Gamma }}}{1}\right)}^{\tfrac{1}{2}}{\left(\displaystyle \frac{{E}_{\mathrm{ph},{\rm{b}}}}{{10}^{-2}\mathrm{eV}}\right)}^{-\tfrac{1}{2}},\end{eqnarray} \tag{ 21 }$$

and obtain a constraint on τ_acc and B in the PWN. When τ_acc is relatively long with ${E}_{\mathrm{cf}}\lt {E}_{l}$ or ${E}_{\mathrm{cf}}\lt {E}_{c}$, the PWN is in Case (ii) or Case (iv). Accordingly, we have

$$\begin{eqnarray}&&\left(\displaystyle \frac{{\tau }_{\mathrm{acc}}}{1\,\mathrm{kyr}}\right){\left(\displaystyle \frac{B}{100\mu {\rm{G}}}\right)}^{\tfrac{3}{2}}=837.0{\left(\displaystyle \frac{{\rm{\Gamma }}}{1}\right)}^{\tfrac{1}{2}}{\left(\displaystyle \frac{{E}_{\mathrm{ph},{\rm{b}}1}}{{10}^{-5}\mathrm{eV}}\right)}^{-\tfrac{1}{2}}.\end{eqnarray} \tag{ 22 }$$

If the spectral shape in the infrared band cannot be observationally determined (e.g., G0.9+0.1, N157B), we can still set a constraint on the range of τ_acc and B,

$$\begin{eqnarray}&&\begin{array}{l}26.5{\left(\displaystyle \frac{{\rm{\Gamma }}}{1}\right)}^{\tfrac{1}{2}}{\left(\displaystyle \frac{{E}_{\mathrm{ph},{\rm{b}}}}{{10}^{-2}\mathrm{eV}}\right)}^{-\tfrac{1}{2}}\leqslant \left(\displaystyle \frac{{\tau }_{\mathrm{acc}}}{1\,\mathrm{kyr}}\right){\left(\displaystyle \frac{B}{100\mu {\rm{G}}}\right)}^{\tfrac{3}{2}}\\ \qquad \qquad \leqslant \,837.0{\left(\displaystyle \frac{{\rm{\Gamma }}}{1}\right)}^{\tfrac{1}{2}}{\left(\displaystyle \frac{{E}_{\mathrm{ph},{\rm{b}}1}}{{10}^{-5}\mathrm{eV}}\right)}^{-\tfrac{1}{2}}.\end{array}\end{eqnarray} \tag{ 23 }$$

In Table 2, we list the values of ${E}_{\mathrm{ph},{\rm{b}}}$ and ${E}_{\mathrm{ph},{\rm{b}}1}$ of the above PWNe. For G0.9+0.1 and N157B, we can only provide the lower limit of ${E}_{\mathrm{ph},{\rm{b}}1}$ due to the insufficient data points. By using the estimated magnetic field strength ${B}_{\mathrm{eq}}$ under the assumption of equipartition between particle and magnetic energies (see Table 2), in Figure 5 we present the corresponding τ_acc (Equations (21) and (22)) or the range of τ_acc (Equation (23)) inferred from the spectral breaks of the PWNe. Empirically, we see that τ_acc of different PWNe are basically within the range confined by Equation (23).

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Table 2.  Parameters of the PWNe

PWNe	Mouse	3C 58	G21.5–0.9	G0.9+0.1	N157B
${E}_{\mathrm{ph},{\rm{b}}}$ [eV]	$9.0\times {10}^{-3}$	⋯	$2.1\times {10}^{-4}$	$5.8\times {10}^{-2}$	$2.2\times {10}^{-2}$
${E}_{\mathrm{ph},{\rm{b}}1,\min }$ [eV]	$3.1\times {10}^{-4}$	$7.3\times {10}^{-5}$	⋯	$5.0\times {10}^{-5}$	$2.1\times {10}^{-5}$
${B}_{\mathrm{eq}}$ [μG]	200a	60b	180c	56d	230e

Notes.

^a K18. ^bReynolds & Aller (1988). ^cSafi-Harb et al. (2001). ^dDubner et al. (2008). ^eWang et al. (2001).

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